# Full text of "History Of The Theory Of Numbers - I"

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```CHAP, xiii]               FACTOB TABLES, LISTS OF PRIMES.                       355
A. Cunningham 5 noted errata in various factor tables.
*J. R. Akerlund85a discussed the determination of primes by a machine.
Gaston Tarry86 would use an auxiliary table (as did Barlow in 1819) to tell by the addition of two entries (< Jp) if a given number <N is divisible by a chosen prime p. For N = 10000, he used the base b = 100, and gave a table showing the numerically least residues of the numbers r<b and the multiples of b for each prime p<6. Then rib+r is divisible by p if the residues of rib and r are equal and of opposite sign. For JV = 100 000, he used & = 60060=2-91-330 and wrote numbers in the form wfc+330g+r, g<90, r<330; or, again, 5 = 20580. Ernest Lebon87 used such tables with the base 30030 = 2-3-5-7-lM3, or its product by 17.
Ernest Lebon,88 J. Deschamps,89 and C. A. Laisant89a discussed the construction of factor tables.
J. C. Morehead90 extended the sieve of Eratosthenes to numbers mak+b (m = l, 2, 3,...) hi any arithmetical progression. The case a = 2, fc = =*= 1, is discussed in detail, with remarks on the construction of a table to serve as a factor table for numbers m-2k=*= 1.
L. L. Dines91 treated the case a = 6, & = =*=!, and the factorization of numbers ra-G*^ 1.
D. N. Lehmer92 gave a factor table to 10 million and listed the errata in the tables by Burckhardt,  Glaisher, Dase,  Base  and Rosenberg, and Kulik's tenth million, and gave references to other (shorter) lists of errata.
E. B. Escott92a listed 94 pairs of consecutive large numbers all of whose prime factors are small.
L. Aubry926 proved that a group of 30 consecutive odd numbers does not
contain more than 15 primes or numbers all of whose prime factors exceed 7.
Cunningham920 listed the numbers of 5 digits with prime factors ^11.
"Messenger Math., 34, 1904-5, 24r-31; 35, 1905-6, 24.
MaNyt Tidsskrift for Mat., Kjobenhavn, 16A, 1905, 97-103.
86Bull. Soc. Philoinathique de Paris, (9), 8, 1906, 174-6, 194-6; 9, 1907, 56-9. Sphinx-Oedipe, Nancy, 1906-7, 39-41. Tablettes des Cotes, Gauthier-ViUars, Paris, 1906. Assoc. frang. avanc. sc., 36, 1907, II, 32-42; 41, 1912, 38-43.
87Comptes Rendus Paris, 151,1905,78. Bull. Amer. Math. Soc., 13,1906-7,74. L'enseignement math., 9, 1907, 185. Bull. Soc. Philomathique de Paris, (9), 8, 1906,168, 270; (9), 10, 1908, 4-9, 66-83; (10), 2, 1910, 171-7. Assoc. franc, avanc. sc., 36, 1907, II, 11-20, 49-55; 37, 1909, 33-6; 41, 1912,44-53; 43,1914, 29-35. Rend. Accad. Lincei, Rome, (5), 15, 1906, I, 439; 26, 1917, I, 401-5. Sphinx-Oedipe, 1908-9, 81, 97. Bull. Sc. Math. fil6m., 12, 1907, 292-3. II Pitagora, Palermo, 13, 1906-7, 81-91 (table serving to factor numbers from 30030 to 510 510). Table de caracteMstiques relatives a le base 2310 des facteurs premiers d'un nombre infdrieur & 30030, Paris, 1906, 32 pp. Comptes Rendus Paris, 159, 1914, 597-9; 160, 1915, 758-760; 162, 1916, 346-8; 163,1916, 259-261; 164, 1917, 482-4.
88Jornal de sciencias math., phys. e nat., acad. sc. Lisbona, (2), 7, 1906, 209-218.
8»Bull. Soc. Philomathique de Paris, (9), 9, 1907, 112-128; 10, 1908, 10-41.
8flaAssoc. franc.., 41, 1912, 32-7.
90Annals of Math., (2), 10, 1908-9, 88-104.                                         MJWd.} pp. 105-115.
92Factor table for the first ten millions, Carnegie Inst. Wash. Pub. No. 105, 1909.
92aQuar. Jour. Math., 41, 1910, 160-7; Tinterme'diaire des math., 11, 1904, 65; Math. Quest. Educ. Times, (2), 7, 1905, 81-5.
"^Sphinx-Oedipe, 6, 1911, 187-8; Problem of Lionnet, Nouv. Ann. Math., (3), 2, 1883, 310.
McMath. Quest. Educ. Times, (2), 21, 1912, 82-3.```